Question

If z = 3cis30°, z3 in rectangular form is:

Answer 1

To express z^3 in rectangular form, convert z from polar form to rectangular form by substituting the given values into the formulas x = r * cos(θ) and y = r * sin(θ). Then, raise each component of z to the power of 3.

Step-by-step explanation:

To express z^3 in rectangular form, we first need to convert z from polar form to rectangular form.

Given:
z = 3cis30°

To convert to rectangular form, we use the formulas:
x = r * cos(θ) and y = r * sin(θ)

Substituting the given values, we get:
x = 3 * cos(30°) and y = 3 * sin(30°)

Simplifying, we have:
x = 3 * √3/2 and y = 3 * 1/2

Therefore, z in rectangular form is:
z = 3 * √3/2 + 3i/2

To find z^3, we simply raise each component to the power of 3:
z^3 = (3 * √3/2 + 3i/2)^3

Simplifying further, we get:
z^3 = (81√3/8 - 27i√3/4)

Answer 2

`\qquad \textit{power of two complex numbers} \\\\\ [\quad r[cos(\theta)+isin(\theta)]\quad ]^n\implies r^n[cos(n\cdot \theta)+isin(n\cdot \theta)] \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ z=3[\cos(30^o)+i\sin(30^o)] \\\\\\ z^3=3^3[\cos(3[30^o])+i\sin(3[30^o])]\implies z^3=27[\cos(90^o)+i\sin(90^o)] \\\\\\ z^3=27(0~~ + ~~i1)\implies z^3=0+27i`

so that gives us the point (0 , 27) on the imaginary plane, and we also know the line goes through the origin, since it'd be a line in component form, and to get the equation of a line we only need two points off of it, let's use those two.

`\stackrel{origin}{(\stackrel{x_1}{0}~,~\stackrel{y_1}{0})}\qquad (\stackrel{x_2}{0}~,~\stackrel{y_2}{27}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{27}-\stackrel{y1}{0}}}{\underset{run} {\underset{x_2}{0}-\underset{x_1}{0}}}\implies \cfrac{27}{0}\implies und efined`

a line with an undefined slope is a vertical line, namely in this case x = 27.